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  <div class="section" id="numpy-matmul">
<h1>numpy.matmul<a class="headerlink" href="#numpy-matmul" title="Permalink to this headline">¶</a></h1>
<dl class="data">
<dt id="numpy.matmul">
<code class="sig-prename descclassname">numpy.</code><code class="sig-name descname">matmul</code><span class="sig-paren">(</span><em class="sig-param">x1</em>, <em class="sig-param">x2</em>, <em class="sig-param">/</em>, <em class="sig-param">out=None</em>, <em class="sig-param">*</em>, <em class="sig-param">casting='same_kind'</em>, <em class="sig-param">order='K'</em>, <em class="sig-param">dtype=None</em>, <em class="sig-param">subok=True</em><span class="optional">[</span>, <em class="sig-param">signature</em>, <em class="sig-param">extobj</em><span class="optional">]</span><span class="sig-paren">)</span><em class="property"> = &lt;ufunc 'matmul'&gt;</em><a class="headerlink" href="#numpy.matmul" title="Permalink to this definition">¶</a></dt>
<dd><p>Matrix product of two arrays.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>x1, x2</strong><span class="classifier">array_like</span></dt><dd><p>Input arrays, scalars not allowed.</p>
</dd>
<dt><strong>out</strong><span class="classifier">ndarray, optional</span></dt><dd><p>A location into which the result is stored. If provided, it must have
a shape that matches the signature <em class="xref py py-obj">(n,k),(k,m)-&gt;(n,m)</em>. If not
provided or None, a freshly-allocated array is returned.</p>
</dd>
<dt><strong>**kwargs</strong></dt><dd><p>For other keyword-only arguments, see the
<a class="reference internal" href="../ufuncs.html#ufuncs-kwargs"><span class="std std-ref">ufunc docs</span></a>.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.16: </span>Now handles ufunc kwargs</p>
</div>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>y</strong><span class="classifier">ndarray</span></dt><dd><p>The matrix product of the inputs.
This is a scalar only when both x1, x2 are 1-d vectors.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl>
<dt><strong>ValueError</strong></dt><dd><p>If the last dimension of <em class="xref py py-obj">a</em> is not the same size as
the second-to-last dimension of <em class="xref py py-obj">b</em>.</p>
<p>If a scalar value is passed in.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.vdot.html#numpy.vdot" title="numpy.vdot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">vdot</span></code></a></dt><dd><p>Complex-conjugating dot product.</p>
</dd>
<dt><a class="reference internal" href="numpy.tensordot.html#numpy.tensordot" title="numpy.tensordot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tensordot</span></code></a></dt><dd><p>Sum products over arbitrary axes.</p>
</dd>
<dt><a class="reference internal" href="numpy.einsum.html#numpy.einsum" title="numpy.einsum"><code class="xref py py-obj docutils literal notranslate"><span class="pre">einsum</span></code></a></dt><dd><p>Einstein summation convention.</p>
</dd>
<dt><a class="reference internal" href="numpy.dot.html#numpy.dot" title="numpy.dot"><code class="xref py py-obj docutils literal notranslate"><span class="pre">dot</span></code></a></dt><dd><p>alternative matrix product with different broadcasting rules.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The behavior depends on the arguments in the following way.</p>
<ul class="simple">
<li><p>If both arguments are 2-D they are multiplied like conventional
matrices.</p></li>
<li><p>If either argument is N-D, N &gt; 2, it is treated as a stack of
matrices residing in the last two indexes and broadcast accordingly.</p></li>
<li><p>If the first argument is 1-D, it is promoted to a matrix by
prepending a 1 to its dimensions. After matrix multiplication
the prepended 1 is removed.</p></li>
<li><p>If the second argument is 1-D, it is promoted to a matrix by
appending a 1 to its dimensions. After matrix multiplication
the appended 1 is removed.</p></li>
</ul>
<p><code class="docutils literal notranslate"><span class="pre">matmul</span></code> differs from <code class="docutils literal notranslate"><span class="pre">dot</span></code> in two important ways:</p>
<ul>
<li><p>Multiplication by scalars is not allowed, use <code class="docutils literal notranslate"><span class="pre">*</span></code> instead.</p></li>
<li><p>Stacks of matrices are broadcast together as if the matrices
were elements, respecting the signature <code class="docutils literal notranslate"><span class="pre">(n,k),(k,m)-&gt;(n,m)</span></code>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">([</span><span class="mi">9</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">([</span><span class="mi">9</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(9, 5, 7, 9, 5, 3)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(9, 5, 7, 3)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># n is 7, k is 4, m is 3</span>
</pre></div>
</div>
</li>
</ul>
<p>The matmul function implements the semantics of the <em class="xref py py-obj">&#64;</em> operator introduced
in Python 3.5 following PEP465.</p>
<p class="rubric">Examples</p>
<p>For 2-D arrays it is the matrix product:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="gp">... </span>              <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="gp">... </span>              <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">array([[4, 1],</span>
<span class="go">       [2, 2]])</span>
</pre></div>
</div>
<p>For 2-D mixed with 1-D, the result is the usual.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="gp">... </span>              <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="go">array([1, 2])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="go">array([1, 2])</span>
</pre></div>
</div>
<p>Broadcasting is conventional for stacks of arrays</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">*</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">*</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(2, 2, 2)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="go">98</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">:]</span> <span class="o">*</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span> <span class="p">,</span> <span class="p">:,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">98</span>
</pre></div>
</div>
<p>Vector, vector returns the scalar inner product, but neither argument
is complex-conjugated:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">([</span><span class="mi">2</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="n">j</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="n">j</span><span class="p">])</span>
<span class="go">(-13+0j)</span>
</pre></div>
</div>
<p>Scalar multiplication raises an error.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="mi">3</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">ValueError</span>: <span class="n">matmul: Input operand 1 does not have enough dimensions ...</span>
</pre></div>
</div>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.10.0.</span></p>
</div>
</dd></dl>

</div>


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